feat(algorithms): add stable bubble sort

Cslib.lean

@@ -1,6 +1,8 @@
 module  -- shake: keep-all
 
+public import Cslib.Algorithms.Lean.BubbleSort.BubbleSort
 public import Cslib.Algorithms.Lean.MergeSort.MergeSort
+public import Cslib.Algorithms.Lean.Sorting
 public import Cslib.Algorithms.Lean.TimeM
 public import Cslib.Computability.Automata.Acceptors.Acceptor
 public import Cslib.Computability.Automata.Acceptors.OmegaAcceptor

Cslib/Algorithms/Lean/BubbleSort/BubbleSort.lean

@@ -0,0 +1,205 @@
+/-
+Copyright (c) 2026 Robert Joseph George. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Robert Joseph George
+-/
+
+module
+
+public import Cslib.Algorithms.Lean.Sorting
+public import Cslib.Algorithms.Lean.TimeM
+public import Mathlib.Data.Nat.Basic
+public import Mathlib.Order.Lattice
+public import Mathlib.Tactic.Attr.Core
+public import Mathlib.Tactic.Finiteness.Attr
+public import Batteries.Data.List.Lemmas
+
+/-!
+# Stable bubble sort on lists
+
+`bubbleSort` returns a `TimeM ℕ (List α)`: the return value is the sorted list, and the time
+component counts comparisons.
+
+Each pass moves one maximal element to the end. Stability comes from the strict swap test: equal
+elements are copied forward in their original order instead of being exchanged.
+-/
+
+@[expose] public section
+
+set_option autoImplicit false
+
+namespace Cslib.Algorithms.Lean.TimeM
+
+variable {α : Type} [LinearOrder α]
+
+/--
+One stable bubbling pass. `bubbleMax candidate xs` returns a prefix and a final element.
+The final element is maximal among `candidate :: xs`.
+
+The pass swaps only when the next value is strictly smaller than the current candidate. Equal values
+are therefore never swapped, which is the local reason the full sort is stable.
+-/
+def bubbleMax : α → List α → TimeM ℕ (List α × α)
+  | candidate, [] => return ([], candidate)
+  | candidate, x :: xs => do
+    ✓ let swap := x < candidate
+    if swap then
+      let result ← bubbleMax candidate xs
+      return (x :: result.1, result.2)
+    else
+      let result ← bubbleMax x xs
+      return (candidate :: result.1, result.2)
+
+/-- A bubbling pass keeps exactly one element out of the prefix. -/
+@[simp, grind =]
+private theorem bubbleMax_length (candidate : α) (xs : List α) :
+    (⟪bubbleMax candidate xs⟫).1.length = xs.length := by
+  induction xs generalizing candidate with
+  | nil => simp [bubbleMax]
+  | cons x xs ih =>
+    simp [bubbleMax]
+    split <;> simp [ih]
+
+/-- A bubbling pass performs one comparison for each element it scans. -/
+@[simp, grind =]
+private theorem bubbleMax_time (candidate : α) (xs : List α) :
+    (bubbleMax candidate xs).time = xs.length := by
+  induction xs generalizing candidate with
+  | nil => simp [bubbleMax]
+  | cons x xs ih =>
+    simp [bubbleMax]
+    split <;> simp [ih, Nat.add_comm]
+
+/-- Sorts a list using stable bubble sort, counting comparisons as time cost. -/
+def bubbleSort : List α → TimeM ℕ (List α)
+  | [] => return []
+  | x :: xs =>
+    let pass := bubbleMax x xs
+    let sortedPrefix := bubbleSort pass.ret.1
+    ⟨sortedPrefix.ret ++ [pass.ret.2], pass.time + sortedPrefix.time⟩
+termination_by xs => xs.length
+decreasing_by
+  simp_wf
+
+section Correctness
+
+/-- The final element produced by a bubbling pass is at least the initial candidate. -/
+@[simp]
+private theorem bubbleMax_candidate_le_output (candidate : α) (xs : List α) :
+    candidate ≤ (⟪bubbleMax candidate xs⟫).2 := by
+  induction xs generalizing candidate with
+  | nil => simp [bubbleMax]
+  | cons x xs ih =>
+    by_cases h : x < candidate
+      <;> simp only [bubbleMax, h, ↓reduceIte, ret_bind, ret_pure]
+      <;> grind
+
+/-- Every element left in the prefix of a bubbling pass is bounded by the final element. -/
+@[simp]
+private theorem bubbleMax_prefix_le_output (candidate : α) (xs : List α) :
+    ∀ z ∈ (⟪bubbleMax candidate xs⟫).1, z ≤ (⟪bubbleMax candidate xs⟫).2 := by
+  induction xs generalizing candidate with
+  | nil => simp [bubbleMax]
+  | cons x xs ih =>
+    by_cases h : x < candidate
+      <;> simp only [bubbleMax, h, ↓reduceIte, ret_bind, ret_pure, List.mem_cons]
+      <;> intro z hz
+      <;> rcases hz with rfl | hz
+    · exact le_trans (le_of_lt h) (bubbleMax_candidate_le_output candidate xs)
+    · exact ih candidate z hz
+    · exact le_trans (le_of_not_gt h) (bubbleMax_candidate_le_output x xs)
+    · exact ih x z hz
+
+/-- A bubbling pass is a permutation of the candidate and the remaining input. -/
+@[simp]
+private theorem bubbleMax_perm (candidate : α) (xs : List α) :
+    ((⟪bubbleMax candidate xs⟫).1 ++ [(⟪bubbleMax candidate xs⟫).2]).Perm
+      (candidate :: xs) := by
+  induction xs generalizing candidate with
+  | nil => simp [bubbleMax]
+  | cons x xs ih =>
+    by_cases h : x < candidate
+    · simpa only [bubbleMax, h, ↓reduceIte, ret_bind, ret_pure, List.cons_append] using
+        (List.Perm.cons x (ih candidate)).trans (List.Perm.swap candidate x xs)
+    · simpa only [bubbleMax, h, ↓reduceIte, ret_bind, ret_pure, List.cons_append] using
+        List.Perm.cons candidate (ih x)
+
+/--
+A bubbling pass is stable. Since equal values are never swapped, filtering by any value gives the
+same subsequence before and after the pass.
+-/
+private theorem bubbleMax_stable (candidate : α) (xs : List α) :
+    StableByValue (candidate :: xs)
+      ((⟪bubbleMax candidate xs⟫).1 ++ [(⟪bubbleMax candidate xs⟫).2]) := by
+  induction xs generalizing candidate with
+  | nil => simp [StableByValue, bubbleMax]
+  | cons x xs ih =>
+    intro y
+    by_cases h : x < candidate
+    · have := ih candidate y
+      by_cases hxy : x = y <;> by_cases hcy : candidate = y <;> grind [bubbleMax]
+    · have := ih x y
+      by_cases hcy : candidate = y <;> grind [bubbleMax]
+
+/-- Bubble sort returns a permutation of its input. -/
+theorem bubbleSort_perm (xs : List α) : (⟪bubbleSort xs⟫).Perm xs := by
+  fun_induction bubbleSort xs with
+  | case1 => simp
+  | case2 x xs pass _ ih =>
+    exact (List.Perm.append_right [pass.ret.2] ih).trans (by simp [pass])
+
+/-- Bubble sort is stable: equal values occur in the same order after sorting. -/
+theorem bubbleSort_stable (xs : List α) : StableByValue xs ⟪bubbleSort xs⟫ := by
+  fun_induction bubbleSort xs with
+  | case1 => simp [StableByValue]
+  | case2 x xs pass _ ih =>
+    intro y
+    simp only [List.filter_append]
+    rw [ih y]
+    simpa [pass, List.filter_append] using bubbleMax_stable x xs y
+
+/--
+Bubble sort returns a sorted list. The recursive call sorts the prefix, and `bubbleMax` proves every
+prefix element is below the final element appended at the end.
+-/
+theorem bubbleSort_sorted (xs : List α) : List.Pairwise (· ≤ ·) ⟪bubbleSort xs⟫ := by
+  fun_induction bubbleSort xs with
+  | case1 => simp
+  | case2 x xs pass _ ih =>
+    change ((bubbleSort pass.ret.1).ret ++ [pass.ret.2]).Pairwise (· ≤ ·)
+    rw [List.pairwise_append]
+    have hprefix := bubbleMax_prefix_le_output x xs
+    have hperm := bubbleSort_perm pass.ret.1
+    grind
+
+/-- Bubble sort is functionally correct. -/
+theorem bubbleSort_correct (xs : List α) : List.Pairwise (· ≤ ·) ⟪bubbleSort xs⟫ ∧
+    (⟪bubbleSort xs⟫).Perm xs ∧ StableByValue xs ⟪bubbleSort xs⟫ := by
+  exact ⟨bubbleSort_sorted xs, bubbleSort_perm xs, bubbleSort_stable xs⟩
+
+end Correctness
+
+section TimeComplexity
+
+/-- Bubble sort performs at most a full bubbling pass for each output position. -/
+theorem bubbleSort_time (xs : List α) :
+    let n := xs.length
+    (bubbleSort xs).time ≤ n * n := by
+  fun_induction bubbleSort xs with
+  | case1 => simp
+  | case2 x xs pass _ ih =>
+    simp only [List.length_cons]
+    change pass.time + (bubbleSort pass.ret.1).time ≤ (xs.length + 1) * (xs.length + 1)
+    have hpass : pass.ret.1.length = xs.length := by simp [pass]
+    calc
+      pass.time + (bubbleSort pass.ret.1).time
+          ≤ pass.time + pass.ret.1.length * pass.ret.1.length := by omega
+      _ = xs.length + pass.ret.1.length * pass.ret.1.length := by simp [pass]
+      _ = xs.length + xs.length * xs.length := by rw [hpass]
+      _ = xs.length * (xs.length + 1) := by simp [Nat.mul_succ, Nat.add_comm]
+      _ ≤ (xs.length + 1) * (xs.length + 1) :=
+        Nat.mul_le_mul_right (xs.length + 1) (Nat.le_succ xs.length)
+
+end TimeComplexity
+
+end Cslib.Algorithms.Lean.TimeM

Cslib/Algorithms/Lean/Sorting.lean

@@ -0,0 +1,28 @@
+/-
+Copyright (c) 2026 Robert Joseph George. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Robert Joseph George
+-/
+
+module
+
+public import Cslib.Init
+
+/-!
+# Sorting utilities
+
+For stable list sorts, filtering the input and output by any value records the relative order of
+the copies of that value.
+-/
+
+@[expose] public section
+
+set_option autoImplicit false
+
+namespace Cslib.Algorithms.Lean
+
+/-- `ys` preserves the order of equal values from `xs`. -/
+abbrev StableByValue {α : Type*} [DecidableEq α] (xs ys : List α) : Prop :=
+  ∀ value, ys.filter (fun x => x = value) = xs.filter (fun x => x = value)
+
+end Cslib.Algorithms.Lean